All Questions
Tagged with ds.dynamical-systems sequences-and-series
11 questions with no upvoted or accepted answers
57
votes
0
answers
3k
views
On the first sequence without triple in arithmetic progression
In this Numberphile video (from 3:36 to 7:41), Neil Sloane explains an amazing sequence:
It is the lexicographically first among the sequences of positive integers without triple in arithmetic ...
9
votes
0
answers
225
views
On the first sequence without collinear triple
Let $u_n$ be the sequence lexicographically first among the sequences of nonnegative integers with graphs without collinear three points (as for $a_n=n^2$ or $b_n=2^n$). It is a variation of that one.
...
8
votes
0
answers
197
views
The condition on $\alpha$ that $\alpha^n$ is convergent modulo 1
We consider numbers $\alpha\in \mathbb{R}$ with $|\alpha|>1$. Is there any result about a characterization of those $\alpha$ so that $\{\alpha^n\}_{n\in \mathbb{N}}$ is convergent modulo 1?
I ...
- 661
7
votes
0
answers
429
views
Dynamics of a curious bijection of $\mathbb N$
The two sequences A48680 and A48679 of the OEIS define two mutually inverse bijections on the set of all strictly positive natural numbers given (for the comfort of the reader) as follows:
Given an ...
- 17.9k
4
votes
0
answers
114
views
Is this pair of coupled sequences known, and what are their properties?
I was examining the following pair of 'coupled' sequences (I don't know the correct terminology):
$a_{n+1}=a_n+b_n+\frac{a_n}{b_n}$
$b_{n+1}=b_n\left(1+\frac{b_n}{a_n}\right)$
Both sequences grow ...
4
votes
0
answers
93
views
Flow of zeros in the shifted exponential generating function?
Given a sequence $a_n$ (of real numbers, described more fully below), one may define the exponential generating function (on the complex plane) as $E(z)=\sum_{n=0}^\infty a_n z^n/n!$. The derivatives $...
- 384
3
votes
0
answers
123
views
Irregularly Intertwined Linear Recursions: Other References?
I was wondering if anyone had run across the following notion of intertwined linear recursions. I'm looking for references, or even a standard name. (I know one source, which is the genesis of this ...
- 47.4k
2
votes
0
answers
210
views
A sum with integer parts
Let $ \mathcal{A} $ be a set of reals such that $ \sum_{a \in \mathcal{A} } \frac{1}{a} = \infty $ and $ \sum_{a \in \mathcal{A} } \frac{1}{a^2} < \infty $. For instance, $ \mathcal{A} = \mathbb{N}^...
- 593
1
vote
0
answers
84
views
Coarse well-distributedness/equidistribution of Pell sequence prefixes
I am interested in the distributedness or "mixing" behavior of certain
linear recurrences modulo powers of $2$.
In particular, consider the Pell sequence (https://oeis.org/A000129),
modulo $...
- 11
1
vote
0
answers
189
views
Vandermonde shift
I'm looking for any known results on a shift operator commutated by a Vandermonde matrix. That is, let
$$T=\begin{bmatrix}0 & 1 & 0 & 0 & \cdots \\
0 & 0 & 1 & 0 & \...
- 384
0
votes
0
answers
42
views
Convergence of a positive sequence controlled by a difference inequality involving quadratic map
I have a sequence $\{x_n\}_{n\ge 0}$ with $x_0>0$, controlled by the difference inequality: $$x_{n+1}\le ax_n^2+b$$ where, $a,b>0$. Had $b$ been $0$ and $a<1$, I would find $x_n\to 0$ as $n\...
